Who is asking: Student
Level: Secondary
Question:
How do you compare to see if a sample standard deviation is different than the population standard deviation? I know how to compare means, but not standard deviations.
Hi Karen,
When testing a hypothesis about the mean mu of a population we take a random sample of size n, find the sample mean m and sample standard deviation s and then calculate
t = (m - mu)/(s/sqrt(n)) (The notation you use might be different.)
A decision is then made about the hypothesis by comparing this number t to number in a table, either a standard normal distribution table or the t-distribution with n-1 degrees of freedom. (The choice of distribution depends on the sample size and which book you use.)
To test a hypothesis about the standard deviation sigma of a population the procedure is similar but the number you calculate is different and so is the table you use. Again you take a random sample of size n and find the sample standard deviation s. The number you calculate this time is
chi-squared = (n - 1) s2/sigma2
To make a decision about your hypothesis you compare this number to a critical number as you do for the means test, but the distribution you use in the chi-squared distribution with n-1 degrees of freedom.
Besides having to use a different table, a chi-squared table, there is another complication with this test. To apply this test you need to ensure that the population distribution is a normal distribution.
I hope this helps,
Penny